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Homesqueezing $\mathbb{R}^n$

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# squeezing $\mathbb{R}^{n}$

Squeezing the vector space $\mathbb{R}^{n}$ in the direction of one coordinate axis, i.e. multiplying a certain component $x_{i}$ of all vectors by a non-zero real number $k$, is a linear transformation of $\mathbb{R}^{n}$.

A concrete example of such squeezing and its results is obtained if we squeeze in $\mathbb{R}^{2}$, i.e. in the Euclidean plane formed by all pairs $(x,\,y)$ of real numbers, every $y$-coordinate by a positive number $k=\frac{b}{a}$ where $a>b>0$. We may look how this procedure acts on the circle

$\displaystyle x^{2}+y^{2}=a^{2}.$ | (1) |

Since all ordinates of this equation are shrinked by the factor $\displaystyle\frac{b}{a}$ which is less than 1, we must must multiply the new $y$ in equation (1) by the inverse number $\displaystyle\frac{a}{b}$ in order to keep the equation satisfied; then the new $y$ no longer denotes the ordinate of the circle, but rather the ordinate of the squeezed circle. Thus, the equation of the squeezed curve is

$x^{2}+\left(\frac{a}{b}\!\cdot\!y\right)^{2}=a^{2}.$ |

Simplifying we first obtain

$x^{2}+\frac{a^{2}y^{2}}{b^{2}}=a^{2},$ |

and dividing all terms by $a^{2}$ yields

$\displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1.$ | (2) |

So the resulting curve is an ellipse with semiaxes $a$ and $b$.

In the picture below, the circle $x^{2}\!+\!y^{2}=a^{2}$ is drawn in red and the ellipse $\displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ in blue. The angle $t$ is the eccentric anomaly at the point $P$ of the ellipse, which has the parametric presentation $x=a\cos{t}$, $y=b\sin{t}$.

## Mathematics Subject Classification

15-00*no label found*15A04

*no label found*

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## Comments

## Point in pstricks

A simple question: How can one highlight a point in a diagram (e.g. in "squeeze R^n" the points P of the ellipse and the corresponding point on the circle)?

Jussi

## Re: Point in pstricks

You just need psdot. I have edited the entry in question accordingly.

(In the case of more than one point, one can use psdots, then list all of the points one wants to label.)

Warren

## Re: Point in pstricks

Thank you, Warren, it is exactly what I had hoped! I myself tried "\pspoint" before, but the system did not like it...

Jussi

## Re: Point in pstricks

\pspoint should work, but the result is just a period-sized dot that will normally get lost amongst the other graphics.